Explore machine learning analysis of chaos and chaos theory's insights into machine learning in this comprehensive lecture by Edward Ott from the University of Maryland. Delve into the use of machine learning to understand chaotic dynamical systems through time series measurements, and discover how chaos theory explains the effectiveness of machine learning in this task. Examine illustrative examples like the Kuramoto-Sivashinsky equation, short-term forecasting of chaos, and the replication of long-term dynamical behavior. Investigate the calculation of Lyapunov exponents, the impact of symmetries on machine learning systems, and the conditions for successful replication of ergodic behavior. Gain insights into generalized synchronization and the echo state property in open-loop configurations, providing a comprehensive understanding of the intersection between machine learning and chaos theory.
Machine Learning Analysis of Chaos and Vice Versa - Edward Ott, University of Maryland
Alan Turing Institute via YouTube
Overview
Syllabus
Intro
THE GENERAL MACHINE LEARNING (ML) SET UP FOR THIS TALK
illustrative Example: The Kuramoto-Sivashinsky (KS) Equation The KS equation is a spatiotemporally chaotic system
Results: Short Term Forecasting of Chaos
Has the long term dynamical behavior of the Kuramoto-Sivashinsky system truly been replicated? • If it has, even after the sensitive dependence implied by chaos causes prediction to fail, the ergodic statistical properties of the autonomous closed-loop reservoir dynamical system should closely resemble those of the true system.
Finding Ergodic Properties of a Dynamical System Purely from a Finite Time Series Data String
Some Tasks that Machine Learning of Ergodic Behavior Might Enable
Lyapunov exponents provide useful information • The expected duration of useful forecasts of a chaotic process can be estimated as the log of the amount of uncertainty in the specification of initial conditions divided by the largest Lyapunov exponent.
Lyapunov Exponents for the Example of the KS Equation
Our Modification of the Lyapunov Exponent Spectrum Remove Red: True KS system two zero
Why the ML System Does Not See the Galilean Symmetry Associated with the Galilean symmetry is a constant of motion of the KS equation
Question: Aside from possible corrections for symmetries, does the short term, closed loop forecasting machine learning system always successfully replicate the long term ergodic behavior of the chaotic process?
EXAMPLE: THE KS EQUATION Prediction succeeds. Replication of long term
WHEN, HOW, AND WHY IS ERGODIC BEHAVIOR REPLICATED? WHEN, HOW, AND WHY CAN REPLICATION FAIL?
Outline of Lu et al. Let A denote the attractor of the unknown dynamical system from which the measurements are taken. Lu et al. show that training can ideally lead to the existence of an invariant set in the dynamics of the closed-loop ML configuration, where is the image of A
GENERALIZED SYNCHRONIZATION AND THE ECHO STATE PROPERTY OF THE OPEN-LOOP CONFIGURATION
Taught by
Alan Turing Institute
